Microsoft PowerPoint

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Lecture 2

Basic Probability Concepts

Statistics for

Civil & Environmental Engineers

Venn Diagram

Venn Diagrams

provide a very useful visual representation of sets and set operations such as the complement, union, intersection, and the other combinations

However, a Venn diagram is not intended to be used in

defining relationships among the probabilities of events

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Statistics for Civil & Environmental Engineers

Venn Diagram(2)

Venn Diagram Example (2)

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Measures of Probability

Probability Axioms

Pr[Ω]= 1 where, Ω: sample space 0≤ Pr[A] ≤1

Pr[A∪B]= Pr[A] + Pr[B]

if A&B are mutually exclusive

Measures of Probability

Addition Rules

From 2nd axiom

Pr[A^c]= 1 – Pr[A]

From 3rd axiom

Pr[A₁+A₂+ ··· + A_k]= Pr[A₁] + Pr[A₂] + ··· + Pr[A_k] if A_iA_j= Ø for any i ≠ j

The general addition rule Pr[A∪B]≡ Pr[A+B]=

((note)) Pr[A∪B∪C]=

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Further Properties of Probability Functions

Property 1: Probability of null event

Pr[Ø]= 0

Property 2: Probability of a contained event Pr[A] ≤ Pr[B] if A

⊂ B

Property 3: Boole’s inequality

Pr[A₁+A₂+ ··· +A_n]≤ Pr[A₁] + Pr[A₂] + ··· + Pr[A_n]

Measures of Probability

Conditional Probability and Multiplication Rule

Definition

Pr[A|B]= Pr[A∩B] / Pr[B]

Pr[A∩B]= Pr[A] Pr[B]

if A&B are independent Applications

Pr[AB]=

Pr[ABC]=

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Measures of Probability

Independence

Definition

Two events defined in a given probability space Aare independentif either the conditional probability of one event equals its marginal probability, or their joint probability equals the product of the marginals

Events A&B independent

• Pr[A|B]= Pr[A] if Pr[B] > 0

or Pr[B|A]= Pr[B] if Pr[A] > 0

• Pr[A∩B]≡ Pr[AB] = Pr[A] Pr[B]

Measures of Probability

Total Probability

Pr[A]=

Venn diagram for the theorem of total probability. Events B_i, i = 1, … , n, are mutually exclusive and exhaustive, and

some of them intersect A

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Bayes’ Theorems

Pr[B_j|A] = ( Pr[A|B_j] Pr[B_j] ) / (∑_iPr[A|B_i] Pr[B_i] )

((example)) FEMA Study(1)

Pr[fail]=

C

₁

C

₂

C

₃

C

₄

Failure

C₁= flood C₂= earthquake C₃= seepage

C₄= normal operation

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((example)) FEMA Study(2)

Pr[C

_i

] Pr[fail│C

_i

] Pr[C

_i

∩fail]

C

₁

0.03 0.10

C

₂

0.02 0.08

C

₂

0.01 0.05

= Pr[fail]

Pr[flood|fail]=